Question

Given the graph of the exponential function, what option is the correct exponential equation for the graph? (The first point is at (3, 9), the second point is at (4, 6)

Answer 1

To find the exponential equation for the graph that passes through the points (3, 9) and (4, 6), we can start with the general form of an exponential function, which is:

\[
y = ab^x
\]

Here, \(a\) is the initial value (the value when \(x=0\)), and \(b\) is the base of the exponential function.

Given the points (3, 9) and (4, 6), we can use these points to set up two equations as follows:

1. From the point (3, 9):
\[
9 = ab^3 \quad \text{(1)}
\]

2. From the point (4, 6):
\[
6 = ab^4 \quad \text{(2)}
\]

Now we have a system of equations. We can divide equation (2) by equation (1):

\[
\frac{6}{9} = \frac{ab^4}{ab^3}
\]

Simplifying the left side gives:

\[
\frac{2}{3} = b
\]

Now that we have found \(b\), we can substitute \(b\) back into either equation to find \(a\). We'll use equation (1):

\[
9 = a\left(\frac{2}{3}\right)^3
\]
\[
9 = a\left(\frac{8}{27}\right)
\]
\[
a = 9 \cdot \frac{27}{8} = \frac{243}{8}
\]

Now we have both \(a\) and \(b\):
\[
a = \frac{243}{8}, \quad b = \frac{2}{3}
\]

Thus, the exponential function that fits the given points is:

\[
y = \frac{243}{8} \left(\frac{2}{3}\right)^x
\]

This is the correct exponential equation for the graph passing through the points (3, 9) and (4, 6).